If sin A = 3/5, find cos A and tan A.

Steps to Solve

1. Apply the Pythagorean Identity

To find cos A, we can use the Pythagorean identity which states that:

$$ \sin^2 A + \cos^2 A = 1 $$

Given that $\sin A = \frac{3}{5}$, we can substitute this into the identity:

$$ \left(\frac{3}{5}\right)^2 + \cos^2 A = 1 $$

2. Calculate sin squared

Calculating $\sin^2 A$:

$$ \left(\frac{3}{5}\right)^2 = \frac{9}{25} $$

3. Substitute and solve for cos A

Now substitute $\sin^2 A$ into the Pythagorean identity:

$$ \frac{9}{25} + \cos^2 A = 1 $$

We then isolate $\cos^2 A$:

$$ \cos^2 A = 1 - \frac{9}{25} $$

4. Calculate 1 in terms of 25

Convert 1 into a fraction with a denominator of 25:

$$ 1 = \frac{25}{25} $$

5. Perform the subtraction

Now we can find $\cos^2 A$:

$$ \cos^2 A = \frac{25}{25} - \frac{9}{25} = \frac{16}{25} $$

6. Find cos A

Take the square root of both sides to find cos A:

$$ \cos A = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5} $$

7. Determine tan A

We know that:

$$ \tan A = \frac{\sin A}{\cos A} $$

Substituting the values:

If $\cos A = \frac{4}{5}$, then:

$$ \tan A = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} $$

If $\cos A = -\frac{4}{5}$, then:

$$ \tan A = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4} $$